Notice that $\binom{m+n}{k}$ is the number of ways of choose $k$ elements between $m+n$ given elements. Then you can think these $m+n$ elements separated in two blocks of $m$ and $n$ elements. So, if you choose
$i$ elements in the block of the $n$ elements and $k-i$ in the block of the $m$ elements, you choose $k$ elements between the $m+n$ given elements.